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PAU Madrid 2024-25 · Distance between parallel lines
Statement. Lines r and s are parallel: r passes through A(0, 0, 6) and s through B(0, 5, 2), and both have direction (3, 4, 0). Find the true length of the distance between r and s. (Adaptation of the original statement: the exam gives the figure; the coordinates are fixed here so it can be built in the editor.)
Step-by-step solution
- Draw both lines with their two projections: r' and r'' (through A' and A'') and s' and s'' (through B' and B'').
- Because they are parallel, their distance is the length of the common perpendicular. A plane perpendicular to their direction cuts them at two points, and the segment joining them is that perpendicular.
- Draw plane pi perpendicular to direction (3, 4, 0) through a point of r. In the plan its horizontal trace comes out perpendicular to r' and s'.
- Find I = r ∩ pi and J = s ∩ pi.
- Segment IJ is the common perpendicular to both lines: its length is the distance. As it lies in a plane perpendicular to them, it is measured in true length.
Result
The distance between r and s is 5 units (I(3, 4, 6) and J(0.6, 5.8, 2)).